On the global solution in the resonance problem of Poincaré

Poincaré formulated a general problem of resonance in the case of a dynamical system which is reducible to one degree of freedom. He introduced the concept of a global solution; in essence, this means that the domain of the solution(s) covers the entire phase plane, comprising regions of libration and circulation. It is the author's opinion that the technique proposed by Poincaré for the construction of a global solution is impractical. Indeed, in §§201 and 211 ofLes méthodes nouvelles de la méchanique céleste, where he describes the passage from shallow resonance to deep resonance, Poincaré asserts an erroneous conclusion. An alternative procedure, which admits secular terms into the determining function and introduces a regularizing function, is outlined. The latter method has been successfully applied to the Ideal Resonance Problem, which is a special case of the more general problem considered by Poincaré, (Garfinkelet al. (1971); Garfinkel (1972).